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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017

ISSN: 2231-5373 http://www.ijmttjournal.org Page 123

Two Kinds of Weakly Berwald Special (α, β) - metrics of Scalar flag curvature

Thippeswamy K.R #1

, Narasimhamurthy S.K #2

1.2 Departme of Mathematics, Kuvempu University, Shankaraghatta - 577451, Shimoga, Karnataka, india.

Abstract: In this paper, we study two important class

of special (α, β)-metrics of scalar flag curvature in

the form of and

(where and

are constants) are of scalar flag curvature. We prove

that these metrics are weak Berwald if and only if

they are Berwald and their flag curvature vanishes.

Further, we show that the metrics are locally

Minkowskian.

Key Words: Finsler metric, metrics, S-

curvature, Weak berwald metric, flag cur-vature,

locally minkowski metric.

1. Introduction

The flag curvature in Finsler geometry is a

natural extension of the sectional curvature in

Riemannian geometry, which is first introduced by

L. Berwald. For a Finsler manifold the flag

curvature is a function of tangent planes

and directions . is said to be of

scalar flag curvature if the flag curvature

is independent of flags

associated with any fixed flagpole . One of the

fundamental problems in Riemann-Finsler geometry

is to study and characterize Finsler metrics of scalar

flag curvature. It is known that every Berwald

metric is a Landsberg metric and also, for any

Berwald metric the -curvature vanishes [2]. In

2003, Shen proved that Randers metrics with

vanishing -curvature and are not

Berwaldian [10]. He also proved that the Bishop-

Gromov volume comparison holds for Finsler

manifolds with vanishing -curvature.

The concept of metrics and its curvature

properties have been studied by various authors

[1],[8],[4],[7]. X. Cheng, X. Mo and Z. Shen

(2003) have obtained the results on the flag

curvature of Finsler metrics of scalar curvature [3].

Yoshikawa, Okubo and M. Matsumoto(2004)

showed the conditions for some metrics to

be weakly- Berwald. Z. Shen and Yildirim (2008)

obtained the necessary and sufficient conditions for

the metric to be projectively flat. They

also obtained the necessary and sufficient conditions

for the metric

to be projectively flat

Finsler metric of constant flag curvature and proved

that, in this case, the flag curvature vanishes

Recently, X. Cheng(2010) has worked on

metrics of scalar flag curvature with

constant S-curvature. The main purpose of the

present paper is to study and characterize the two

important class of weakly-Berwald metrics

and

(where and are constants) are of scalar flag

curvature The terminologies and notations are

referred to [5][2].

2. Preliminaries

Let be an n-dimensional manifold and

denote the tangent bundle of .

A Finsler metric on is a functions

with the following properties:

a)

b)

Minkowskian norm

The pair is called Finsler manifold;

Let be a Finsler manifold and

(2.1)

For a vector induces an inner

product on as follows

where and

Further, the Cartan torsion C and the mean Cartan

torsion are defined as follows[2]:

http://www.ijmttjournal.org/

International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017

ISSN: 2231-5373 http://www.ijmttjournal.org Page 124

(2.2)

Where

(2.3)

(2.4)

Where

Let be a Riemannian metric and

be a 1-form on an n-dimentional

manifold . The norm

,

Let be a C∞ positive function on an open

interval satisfying the following conditions

– –

Then the functions is a Finsler metric

if and only if . Such Finsler metrics are

called metrics.

Let denote the horizontal covariant

derivative with respect to and , respectively.

Let

(2.5)

. (2.6)

For a positive function on

and a number let

–

Where

–

The geodesic of a Finsler metric is

characterized by the following system of second

order ordinary differential equations:

Where

is called as geodesic coefficients of F. For an

metric using a

Maple program, we can get the following [5];

(2.8)

Where denote the spray coefficients of .

We shall denote

and .

Furthermore, let

By a direct computation, we can obtain a formula for

the mean Cartan torsion of metrics as

follows[6]:

– . (2.9)

According to Diecke’s theorem, a Finsler metric is

Riemannian if and only if the mean Cartan torsion

vanishes, . Clearly, an metric

is Riemannian if and only if

(see [6]).

For a Finsler metric on a manifold ,

the Riemann curvature is

defined by

.

Let

http://www.ijmttjournal.org/

International Journal of Mathematics Trends and Technology (IJMTT) – Volume 44 Number 3 April 2017

ISSN: 2231-5373 http://www.ijmttjournal.org Page 125

For a flag where , the

flag curvature of is defined by

, (2.10)

where .

We say that Finlser metric is of scalar flag

curvature if the flag curvature is

independent of the flag By the definition , is of

scalar flag curvature if and only if in a

standard local coordinate system,

, (2.11)

where ([2]).

The Schur Lemma in Finsler geometry tell us that, in

dimension , if is of isotropic flag curvature ,

then it is of constant flag curvature , =

constant.

The Berwald curvature

and mean Berwald curvature

are defined respectively by

A Finsler metric F is called weak Berwald metric if

the mean Berwald curvature vanishes, i.e.,

A Finsler metric is said to be of isotropic

mean Berwald curvature if

where is a scalar function on the manifold

.

Theorem 2.1. [9] For special metric

and

(where are constants) on an n-

dimensional manifold . Then the following are

equivalent:

(a) is of isotropic -curvature,

(b) is of isotropic mean Berwald curvature

(c) is a killing 1-form with constant length with

respect to , i.e., (d) -curvature vanishes , (e) is a weak Berwald metric, where is scalar function on the manifold . Note that, the discussion in [9] doesn’t involve

whether or not is Berwald metric. By the

definitions, Berwald metrics must be weak Berwald

metrics but the converse is not true.

For this observation, we further study the metrics

and

(where are constants).

For a Finsler metric on and n-

dimensional manifold , the Busemann-Hausdorff

volume form is

given by

,

denotes the euclidean volume in . The -

curvature is given by

(2.12)

Clearly, the mean Berwald curvature

can be characterized by use of -curvature as

follows:

A Finsler metric F is said to be of isotropic S-

curvature if where

is a scalar function on the manifold . -

curvature is closely related to the flag curvature.

Shen, Mo and cheng proved the following important

result.

Theorem 2.2. [3] Let be an n-dimensional

Finsler manifold of scalar flag curvatur